Elsewhere, in the south of the country, attackers ambushed a Coptic bishop but he escaped unharmed. Both attacks on Christians in the south and assaults on security personnel in the Sinai have surged in the aftermath of the July 3 overthrow of the country's Islamist president, Mohammed Morsi.
In the deadlier of the Sinai attacks, assailants ambushed the policemen on their way to work in the provincial capital of el-Arish, wounding a third officer and speeding away by car, a security official said.
Gunmen also fired on army checkpoints in el-Arish and nearby town of Sheikh Zuweyid, wounding a total of five soldiers and a civilian, a military official said. One of the wounded soldiers later died in a military hospital, he added. Meanwhile, Egypt's official news agency MENA said gunmen opened fire on the el-Arish post office late Sunday, wounding a 14-year-old girl.
The officials spoke on condition of anonymity because they were not authorized to brief reporters.
The army says militants have killed more than 100 policemen and soldiers since Morsi's ouster, mostly in northern Sinai. The deadliest attack took place Aug. 19 when unidentified gunmen pulled 25 police conscripts off minibuses and shot them dead on the side of a highway.
In response the military launched its largest offensive in the area against militants in years. Militant statements say that civilians including children were killed during the sweep. The security forces have not commented on those allegations.
Few journalists have direct access to what is happening in the area because of security restrictions. One local journalist who wrote on his Facebook page that the army had hit a mosque, harmed civilians and was relocating families, Ahmed Abu-Draa, is being tried in a military court on charges of spreading false information about the army's operations there.
Late on Sunday, a previously unknown Sinai-based militant group posted its first video message on a militant website, claiming that it had killed an "apostate" and an army colonel in a drive-by attack.
The group, which calls itself the al-Nusra Battalion, said it is attached to the Furqan Brigades that claimed responsibility for an attack in August targeting a carrier ship in the strategic Suez Canal. Egyptian authorities said they foiled the attack.
Al-Nusra's short video showed a gunman shooting a man wearing a traditional Arab robe, identified by subtitles as an "apostate." It also showed men in a car firing at a passing vehicle, claiming all three occupants of the car were killed, including a man it identified as Air Defense Col. Mohammed el-Kumi.
The authenticity of the video could not be verified. It was posted on a website that usually carries al-Qaida and other militant groups' statements.
Violence against Christians, who make up more than 10 percent of Egypt's population of 90 million, has also spiked in the aftermath of Morsi's overthrow. Officials described the attacks as a backlash against Christians from Islamist militants and other Morsi supporters who blame the minority group of conspiring to remove him.
In the latest incident, the bishop of the southern province of Minya came under fire from unknown gunmen as he drove into the village of el-Serw, a local activist said.
Ezzat Ibrahim said Bishop Macarious got out of the car and ran into the house of a Coptic Christian for cover, until the gunmen, who were on motorcycles, fled. The bishop was unharmed.
A security official said the bishop was on a visit to resolve ongoing tension in the village, home to some former militants and hard-line Islamists, over a house used by the local Christians to pray. The official was speaking on condition of anonymity because he was not authorized to speak to the media.
Christians, who are heavily restricted in building churches, often use homes for worship. Disputes over the issue often boil over into violence in rural Egypt, and Minya has been especially hit with unrest following Morsi's removal.
The pro-Morsi camp denies it is behind the violence, and says peaceful protests will continue against the military.
Maamoun Youssef contributed reporting from Cairo.